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arXiv · 2007.13176

Signed Euler-Mahonian identities

Sen-Peng Eu·Zhicong Lin·Yuan-Hsun Lo
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Abstract

A relationship between signed Eulerian polynomials and the classical Eulerian polynomials on $\mathfrak{S}_n$ was given by D\'{e}sarm\'{e}nien and Foata in 1992, and a refined version, called signed Euler-Mahonian identity, together with a bijective proof were proposed by Wachs in the same year. By generalizing this bijection, in this paper we extend the above results to the Coxeter groups of types $B_n$, $D_n$, and the complex reflection group $G(r,1,n)$, where the `sign' is taken to be any one-dimensional character. Some obtained identities can be further restricted on some particular set of permutations. We also derive some new interesting sign-balance polynomials for types $B_n$ and $D_n$.

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Sen-Peng Eu, Zhicong Lin, Yuan-Hsun Lo. 2020-07-26. Signed Euler-Mahonian identities. https://arxiv.org/abs/2007.13176

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